The geometry of chaotic attractors can be complex and difficult to describe. It is therefore useful to have quantitative characterizations of such geometrical objects. Perhaps the most basic such characterization is the 'dimension' of the attractor. However, here the word dimension is somewhat ambiguous, and it can have several meanings. Another complication is that the notion of dimension can be extended to incorporate singular properties of the density with which a typical orbit visits different parts of the attractor; i.e., the dimension can depend on the 'natural measure' (defined below) on the attractor and not solely on the geometry of the attractor. General references on the topic of this article are Ott (2002), Ruelle (1989) and Paladin and Vulpiani (1987) and Tel and Gruiz (2006).

The Box Counting Dimension

If the attractor exists in a d-dimensional phase space (d is necessarily an integer), then \(N(\epsilon)\) can be defined with respect to a d-dimensional rectangular grid of grid size \(\epsilon \ .\) Specifically, \(N(\epsilon)\) is the number of d-dimensional cubes of edge length \(\epsilon \) from the grid that are needed to cover the attractor. Furthermore, it is assumed that the limit in (1) exists and does not depend on how the grid is chosen. (It is generally believed that this assumption is true for chaotic attractors that occur in typical situations.) According to Eq. (1), the number of cubes necessary to cover the attractor scales as \( N(\epsilon )\sim \epsilon \) -D0. (We also note that D0 is sometimes also simply called 'the fractal dimension', although this terminology is somewhat ambiguous as there are different definitions of the dimension of a fractal [e.g., see the section below on The Renyi Dimension]). In practice, determination of \(D_0\) from a numerical experiment would proceed as follows (Russell et al., 1980). (i) Evolve an orbit for a long time collecting a list of the locations \(x_j\) of many orbit points equally spaced in time on the attractor. (In creating an orbit, one typically starts from an arbitrary point not on the attractor, and thus the initial points from the transient period, while the orbit is not extremely near the attractor, should be disregarded.) (ii) Pick several small \(\epsilon\) values \(\epsilon _i=K\delta ^i(i=i_{\min}\ ,\) \(i_{\min} +1\ ,\) \(i_{\min} +2,\ldots , i_{\max}\ ,\) and \(0<\delta <1)\ ;\) i.e., the intervals \(\log \epsilon _i-\log \epsilon _{i+1}=-\log \delta\) are equally spaced. (iii) For each \(i\ ,\) calculate \(\bar N(\epsilon _i)\ ,\) the number of d-dimensional cubes of edge length \(\epsilon _i\) that contain at least one orbit point. (iv) Regarding \(\bar N(\epsilon _i)\) as an estimate of \(N(\epsilon _i)\ ,\) plot \(\log \bar N(\epsilon _i)\) versus \(\log (1/\epsilon _i)\ .\) (v) If all goes well this plot will be well fit by a straight line for small enough \(\epsilon _i\ ;\) then take the slope of this straight line as an estimate of \(D_0\) for the attractor.

There are several potential problems with the above procedure. One is that, as \(\epsilon \) is made small, the number of orbit points in the cubes can be very different, with a relatively small minority of cubes having the vast majority of orbit points, while most of the cubes have few orbit points. In fact, there may be a large number of empty boxes that would become filled if the long orbit in step (i) were made still longer. Furthermore, no matter how long the orbit is, if one makes \( \epsilon \) small enough, this will always be a problem. Thus the estimate \(\bar N(\epsilon _i)\) of \(N(\epsilon _i)\) is usually too small, leading the above procedure to underestimate \(D_0\) (possibly only a little if the smallest \(\epsilon _i\) is small enough and the orbit long enough).

A more fundamental objection to the use of \(D_0\) is that we probably should really be more interested in the cubes in which the orbit spends more time, while the definition of \( D_0\ ,\) Eq. (1), makes no distinction between cubes needed to cover the attractor [i.e., \(D_0\) reflects the attractor geometry but not its natural measure (see below)].

Figure 1: Left panel: Chaotic attractor of a driven anharmonic oscillator on the location-position plane of a stroboscopic map taken with the period of the driving. Right panel: Natural measure on the same chaotic attractor. Lighter colors indicate higher local values of the distribution. Both the attractor and the natural measure are fractal. (From T. Tel, M. Gruiz, Chaotic Dynamics, An Introduction Based on Classical Mechanics, Cambridge University Press, 2006, with permission.)

The Natural Measure

The natural measure associated with a chaotic attractor gives the fraction of the time that the long orbit on the attractor spends in any given region of state space. In particular, let \(C\) be a cube in the phase space of the system; let \(x_0\) be an initial condition in the basin of attraction of a chaotic attractor A; let \(x(t,x_0)\) be the orbit originating from the initial condition \(x_0\ ;\) and let \(\mu (C;x_0, T)\) be the fraction of time that \(x(t,x_0)\) spends in \(C\) in the time interval \(0\leq t\leq T\ ,\) and assume that the limit
\[
\mu (C; x_0)=\lim _{T\rightarrow \infty}\mu (C; x_0,T)
\]
exists. If \(\mu (C;x_0)\) takes on the same value for almost all \(x_0\) with respect to Lebesgue measure in the basin of A, then we call this common value, denoted \(\mu (C)\ ,\) the natural measure of \(C\ ,\) i.e., the set of \(x_0\) for which \(\mu (C;x_0)\neq
\mu (C)\) 'has zero d-dimensional volume'.

The Renyi Dimension, \(D_q\)

The Renyi dimension (also called the 'generalized dimension') takes into account the frequency with which cubes are visited via weighting them according to their natural measure. The strength of this weighting is given by an index q. For q>0, the larger q is the stronger the relative weighting of the higher measure boxes. Again consider a partition of the phase space by an \(\epsilon\)-grid, and for each \(\epsilon\) evaluate \(\mu (C_j)\ ,\) the natural measure of the jth \(\epsilon\)-cube \(C_j\) needed to cover the attractor. The order q Renyi dimension of the attractor is (Renyi, 1970; Balatoni and Renyi, 1956; Grassberger and Procaccia, 1983; Hentschel and Procaccia, 1983)
\[\tag{2}
D_q=\lim _{\epsilon \rightarrow 0} \frac{1}{1-q}\frac{\log \left\{ \sum _j[(\mu (C_j))]^q\right\}} {\ell n(1/\epsilon )} .
\]

Note that for q=0, Eq. (2) yields Eq. (1). \(D_q\) can be numerically evaluated in a manner that is similar to that for the box-counting dimension D0, but for q>0 the evaluation will tend to be more accurate, particularly for larger q (assuming an equal investment in computational cost for different q values). An important property of \(D_q\) is that it is a nonincreasing function of q,
\[\tag{3}
D_{q1}\geq D_{q2}\ \ {\rm if}\ \ q_1<q_2 .
\]

Special interest has attached to the values q=1 and q=2, where \(D_1\) has been called the 'information dimension', and \(D_2\) has been called the 'correlation dimension'. The information dimension is obtained from (2) by taking the limit \(q\rightarrow 1\) and applying L'Hospital's rule
\[\tag{4}
D_1=\lim _{\epsilon \rightarrow 0}\frac{\sum _j\mu (C_j)\log [\mu (C_j)]}{\ell n\epsilon } \ .
\]

where \(x_i\) are orbit points as generated in step (i) of our description of how to compute D0. Then plotting \(\log S(\epsilon )\) versus \(\log \epsilon \ ,\) \(D_2\) is estimated as the slope of a straight line fitted to this data for small \(\epsilon \ .\)

Another characterization of the natural measure of chaotic attractors that is essentially equivalent to \(D_q\) involves looking at subsets of the attractor consisting of all points \(x\) such that \(\mu (B_\epsilon (x))\sim \epsilon ^\alpha \ ,\) where \(\alpha \) is 'the singularity index' associated with the point \(x\ .\) Here \(B_\epsilon (x)\) is the d-dimensional phase-space ball centered at the point \(x\ .\) The box-counting dimension of the set of attractor points with singularity index \(\alpha \) is commonly denoted \(f(\alpha )\ ,\) and there exists a transformation from the function of \(\alpha \) given by \(f(\alpha )\) to the function of \(q\) given by \(D_q\) (Grassberger, 1985; Halsey et al., 1986; Paladin and Vulpiani, 1987; Beck and Schlögl 1993; Chapter 9 of Ott, 2002). See entry on Multifractals.

Lyapunov Dimension and the Kaplan-Yorke Conjecture

Kaplan and Yorke (1979) introduced a quantity defined in terms of the Lyapunov exponents \(\lambda _\ell \) \((\ell =1,2,\ldots ,d)\) where the subscript labeling of the \(\lambda _\ell \) is chosen so that \(\lambda _1\geq \lambda _2\geq \ldots \geq \lambda _d\) (i.e., the Lyapunov exponents are arranged in 'size places'). The quantity Kaplan and Yorke introduced, is commonly called the 'Lyapunov dimension' and is given by
\[\tag{6}
D_L=k+\frac{\lambda _1+\lambda _2+\ldots +\lambda _k}{|\lambda _{k+1}|}
\]

where \( k\) is the maximum value of \(i\) such that \(\xi _i=\lambda _1+\ldots +\lambda _i>0\ .\) See the figure for a schematic illustration of Eq. (6). The Kaplan-Yorke conjecture states that
\[\tag{7}
D_1=D_L
\]

for 'typical' systems.

Figure 2: Schematic illustration of Eq. (6).

That is, the information dimension is equal to the Lyapunov dimension. This relationship is remarkable in that it relates dynamics (Lyapunov exponents) to attractor geometry and natural measure \((D_1)\ .\) The restriction to 'typical' systems is necessary because it is not hard to construct examples where (7) is violated. But the claim is that these examples are pathological in that the slightest arbitrary change of the system restores the applicability of (7) and that such violations have 'zero probability' of occurring in practice. While the conjecture has so far defied general proof, it has been proved in some cases. In particular, see Young (1982) and Ledrappier and Young (1988).

The formula for the Kaplan-Yorke dimension is particularly simple in the often encountered case of a chaotic \( (\lambda _1 >0)\ ,\) two-dimensional, area contracting \((\lambda _1+\lambda _2<0)\) map, in which case \(D_L=1+(\lambda _1/|\lambda _2|)\ .\) In addition, there also exists extensions of the Kaplan-Yorke conjecture to the case of nonattracting chaotic sets. In that case, \(D_L\) is given in terms of the Lyapunov exponents and the exponential decay time for orbits to leave the vicinity of the nonattracting chaotic set (Kantz and Grassberger, 1985; Hsu et al., 1988; Hunt et al. 1996). See entry on Chaotic Transients.